# Embedded Cobordism Categories and Spaces of Submanifolds

@article{RandalWilliams2010EmbeddedCC,
title={Embedded Cobordism Categories and Spaces of Submanifolds},
author={Oscar Randal-Williams},
journal={International Mathematics Research Notices},
year={2010},
volume={2011},
pages={572-608}
}
• O. Randal-Williams
• Published 13 December 2009
• Mathematics
• International Mathematics Research Notices
Galatius, Madsen, Tillmann, and Weiss [7] have identified the homotopy type of the classifying space of the cobordism category with objects (d −1)-dimensional manifolds embedded in ℝ ∞ . In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius in [8], to identify the homotopy type of the cobordism category with objects (d −1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle…
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## References

SHOWING 1-10 OF 18 REFERENCES
Geometric cobordism categories
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections
The homotopy type of the cobordism category
• Mathematics
• 2006
The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main
Monoids of moduli spaces of manifolds
• Mathematics
• 2010
We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth
STABLE CHARACTERISTIC CLASSES OF SMOOTH MANIFOLD BUNDLES
Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union t BSOn of classifying spaces of special orthogonal groups SOn with n = 0,1, . . .. A
Embeddings from the point of view of immersion theory
• Philosophy
• 1999
Let M and N be smooth manifolds. For an open V ⊂ M let emb(V,N) be the space of embeddings from V to N . By the results of Goodwillie [4], [5], [6] and Goodwillie–Klein [7], the cofunctor V 7→
Homotopy Limits, Completions and Localizations
• Mathematics
• 1987
Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotent
Configuration-spaces and iterated loop-spaces
The object of this paper is to prove a theorem relating "configurationspaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the
Embeddings from the point of view of immersion theory: Part II
Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V |-->
A parametrized index theorem for the algebraicK-theory Euler class
• Mathematics
• 2003
A Riemann–Roch theorem is a theorem which asserts that some algebraically defined wrong–way map in K –theory agrees or is compatible with a topologically defined one [BFM]. Bismut and Lott [BiLo]